If you watch the stars a lot, you'll notice their location shifts a little each night as the Earth orbits the Sun. There's approximately 360 nights a year, so you could say the amount that each star shifts per night is some unit about 1/360 th of a complete rotation. The Babylonians built a base 60 numbering system based on tracking the stars, hence the 60 minutes per degree and the 60 seconds per minute (and the approximation of 360 days per year - it would be hard to build a numbering system from 365).

The stars also seem to rotate during the night, so the only way you can really use them for navigation is to know what time it is. It only makes sense to make your time units compatible with your position units, hence the 60 minutes per hour, the 60 seconds per minute. Of course, it only takes one day for the Earth to rotate, so the length of the day winds up being 24 hours instead of something more compatible with a base 60 numbering system, but it works well enough.

Edit: Actually, that doesn't really explain why 24 and not some other number since the length of a second didn't have to be the current length. Ideally, the day's segments would be easy to fit back into a circle. The easiest angles are angles like pi/2, pi/3, pi/4, pi/6. pi/12 isn't that much harder than pi/6, and I guess they felt pi/6 was just too long for one hour.

If it was like you're saying (i.e. that one day is 23 hours 56 minutes), then half the year we would have daylight during ordinary nighttime...

What we want is that the sun should be at its highest point in the sky about the same time (12.00) every day. That's why we make sure there is 24 hours between two such events.

(Ok, then we could have day-light saving time and so on, but that's really another story. Also, one hour is not defined as 1/24 of a solar day anymore (it's defined as a certain amount of oscillations in a cesium atom), but the deviation is really tiny, and it's enough to put in some extra second now and then.)

Actually the days are getting longer. The moon's gravity is slowly tugging at the earth, slowing its rotation ever so slightly each year. Billions of years ago the days(rotation of the earth) was only 18 hours long. Eventually the earth will stop spinning altogether but by that time the sun will have long burned out.